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Question
Find the inverse (if it exists) of the following:
`[(5, 1, 1),(1, 5, 1),(1, 1, 5)]`
Solution
A = `[(5, 1, 1),(1, 5, 1),(1, 1, 5)]`
|A| = 5(25 – 1) – (5 – 1) + 1(1 – 5)
= 5(24) – 1(4) + 1(– 4)
= 120 – 4 – 4
= 122 ≠ 0.A–1 exists
adj A = `[(+|(5, 1),(1, 5)|, -|(1, 1),(1, 5)|, +|(1, 5),(1, 1)|),(-|(1, 1),(1, 5)|, +|(5, 1),(1, 5)|, -|(5, 1),(1, 1)|),(+|(1, 1),(5, 1)|, -|(5, 1),(1, 1)|, +|(5, 1),(1, 5)|)]^"T"`
= `[(+(25 - 1), (5 - 1), +(1 - 5)),(-(5 - 1), +(25 - 1), -(5 - 1)),(+(1 - 5), -(5 - 1), +(25 - 1))]^"T"`
= `[(24, -4, -4),(-4, 24, -4),(4, -4, 24)]^"T"`
adj A = `[(24, -4, -4),(-4, 24, -4),(4, -4, 24)]`
`"A"^-1 = 1/|"A"|`
adj A = `1/112 [(24, -4, -4),(-4, 24, -4),(4, -4, 24)]`
= `1/28 [(6, -1, -1),(-1, 6, -1),(-1, -1, 6)]`
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